DYNAMICAL SYSTEMS 1 First semester : For first-year Master’s degree students
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Date
2025
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Faculty of Sciences
Abstract
This course book serves as an introduction to dynamical systems, focusing on ordinary
differential equations (ODEs), their stability, periodic solutions, and bifurcations.
It is aimed at students in mathematics, particularly those in the first year of
their Master’s degree.
The book is structured into five main chapters, each addressing important aspects of
dynamical systems. The first chapter, Preliminaries of Ordinary Differential
Equations, begins with a review of differential systems, followed by their classification,
and linear differential systems, including homogeneous and nonhomogeneous
cases.
The second chapter, General Theory of ODEs, introduces the fundamental aspects
of ordinary differential equations, initial value problems, and solutions. It
includes key existence and uniqueness theorems, different proof methods, and examples.
Additionally, the continuation of solutions and maximal intervals of existence
are discussed in detail.
The third chapter, Stability in Linear and Nonlinear Systems, explores the
concept of stability, starting with linear systems and extending to nonlinear systems.
It covers important tools such as Lyapunov’s method and the analysis of conservative
and dissipative systems, which are fundamental for understanding system behavior
over time.
In the fourth chapter, Periodic Solutions and Their Stability, we delve into
the nature of periodic solutions, limit cycles, and their stability. Concepts such
as Poincar´e maps, Bendixson’s and Dulac’s criteria, and the Poincar´e-Bendixson
theorem are explored, offering a deep understanding of the behavior of dynamical
systems in the long term.
Finally, the fifth chapter, Introduction to Local Bifurcations, introduces the
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concept of bifurcation, focusing on one-dimensional and two-dimensional systems.
Key bifurcations, including saddle-node, pitchfork, transcritical, and Hopf bifurcations,
are explored, laying the foundation for further study and applications in
complex systems.
In this book, my objective was to collect the most important definitions, information,
and tools from the most significant references such as [[12], [14], [15], [10], [5]],
and the references therein, to help students focus on the essential notions of dynamical
systems. Throughout the book, a formal and clear approach is taken, with
numerous examples and illustrations. The topics covered are foundational for the
study of dynamical systems and provide the necessary tools to tackle more advanced
subjects in mathematical modeling, control theory, and applied mathematics.
It is my hope that this book serves as a learning resource for students and a reference
for those seeking to deepen their understanding of dynamical systems